Question: Two mathematicians were both born in the last 500 years. Each lives (or will live) to be 100 years old, then dies. Each mathematician is equally likely to be born at any point during those 500 years. What is the probability that they were contemporaries for any length of time?
Answer: Call the mathematicians Karl and Johann. Let the $x$ axis represent the number of years ago Karl was born, and the $y$ axis represent the number of years ago Johann was born.

[asy]
draw((0,0)--(100,0), Arrow);
draw((0,0)--(0,100), Arrow);
label("0", (0,0), SW);
label("100", (0,20), W);
label("400", (100,80), E);
label("100", (20,0), S);
label("500", (100,0), S);
label("500", (0,100), W);
fill((0,0)--(100,100)--(100,80)--(20,0)--cycle, gray(.7));
fill((0,0)--(100,100)--(80,100)--(0,20)--cycle, gray(.7));
[/asy]

The shaded region represents the years both mathematicians would have been alive. For example, if Karl was born 200 years ago, Johann could have been born anywhere between 300 and 100 years ago. Let 500 years equal one unit.  Then, we can calculate the area of the shaded region as the area of the entire square minus the areas of the two unshaded triangles.  This will be equal to $2\cdot \frac{1}{2} \cdot \frac{4}{5} \cdot \frac{4}{5}=\frac{16}{25}$.  So, the area of the shaded region is $1-\frac{16}{25}=\frac{9}{25}$.  Since the area of the square is 1, this is also the probability that Karl and Johann were contemporaries. The answer, then, is $\boxed{\frac{9}{25}}$.